psubclsubN - Mathbox for Norm Megill

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Theorem psubclsubN 37954

Description: A closed projective subspace is a projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
psubclsub.s	⊢ 𝑆 = (PSubSp‘𝐾)
psubclsub.c	⊢ 𝐶 = (PSubCl‘𝐾)

**Assertion**
Ref	Expression
psubclsubN	⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ 𝑆)

**Proof of Theorem psubclsubN**
Step	Hyp	Ref	Expression
1		eqid 2738	. . 3 ⊢ (⊥_𝑃‘𝐾) = (⊥_𝑃‘𝐾)
2		psubclsub.c	. . 3 ⊢ 𝐶 = (PSubCl‘𝐾)
3	1, 2	psubcli2N 37953	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → ((⊥_𝑃‘𝐾)‘((⊥_𝑃‘𝐾)‘𝑋)) = 𝑋)
4		eqid 2738	. . . . . . 7 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
5	4, 1, 2	psubcliN 37952	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → (𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥_𝑃‘𝐾)‘((⊥_𝑃‘𝐾)‘𝑋)) = 𝑋))
6	5	simpld 495	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → 𝑋 ⊆ (Atoms‘𝐾))
7		psubclsub.s	. . . . . 6 ⊢ 𝑆 = (PSubSp‘𝐾)
8	4, 7, 1	polsubN 37921	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾)) → ((⊥_𝑃‘𝐾)‘𝑋) ∈ 𝑆)
9	6, 8	syldan 591	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → ((⊥_𝑃‘𝐾)‘𝑋) ∈ 𝑆)
10	4, 7	psubssat 37768	. . . 4 ⊢ ((𝐾 ∈ HL ∧ ((⊥_𝑃‘𝐾)‘𝑋) ∈ 𝑆) → ((⊥_𝑃‘𝐾)‘𝑋) ⊆ (Atoms‘𝐾))
11	9, 10	syldan 591	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → ((⊥_𝑃‘𝐾)‘𝑋) ⊆ (Atoms‘𝐾))
12	4, 7, 1	polsubN 37921	. . 3 ⊢ ((𝐾 ∈ HL ∧ ((⊥_𝑃‘𝐾)‘𝑋) ⊆ (Atoms‘𝐾)) → ((⊥_𝑃‘𝐾)‘((⊥_𝑃‘𝐾)‘𝑋)) ∈ 𝑆)
13	11, 12	syldan 591	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → ((⊥_𝑃‘𝐾)‘((⊥_𝑃‘𝐾)‘𝑋)) ∈ 𝑆)
14	3, 13	eqeltrrd 2840	1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ⊆ wss 3887 ‘cfv 6433 Atomscatm 37277 HLchlt 37364 PSubSpcpsubsp 37510 ⊥_𝑃cpolN 37916 PSubClcpscN 37948

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-riotaBAD 36967

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-undef 8089 df-proset 18013 df-poset 18031 df-lub 18064 df-glb 18065 df-join 18066 df-meet 18067 df-p1 18144 df-lat 18150 df-clat 18217 df-oposet 37190 df-ol 37192 df-oml 37193 df-ats 37281 df-atl 37312 df-cvlat 37336 df-hlat 37365 df-psubsp 37517 df-pmap 37518 df-polarityN 37917 df-psubclN 37949

This theorem is referenced by: pclfinclN 37964

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